"Nuts" is slang for "testicles". He's corrupting everyone's speech to say he's putting his testicles on their chin. It's immature humor.
noun [ plural ]
offensive for testicle
Notice that the colleague (Key) is serious and concerned about the behavior of Vince (Peele), but Vince responds in an unexpected, immature way:
Colleague: Vince, I'm worried about you.
Vince: ... I was worried about these nuts on your chin.
Colleague: Vince, seriously.
Vince: These nuts were seriously on your chin.
"These nuts", often spoken as "deez nuts", was a fairly popular punchline for immature jokes a few years ago. There's even an entry on knowyourmeme.com:
Deez Nuts is the punchline of a conversational joke that involves asking someone a vaguely-phrased question to solicit a follow-up question in response, typically in the form of one of the five Ws, before yelling out the said phrase in an obnoxious manner. While originally introduced as a skit track on Dr. Dre's 1992 rap album Chronic, the joke saw a huge resurgence after it was featured in a short comedic sketch by Internet comedian WelvenDaGreat in early 2015.
It's not as widely used these days, but I imagine you'll still hear it in schoolyard jokes.
I don't know if "these nuts on your chin" (the whole phrase) was the comedians' own creation, or a well-known punchline like "deez nuts", but I can imagine that the idea would be universally understood for immature comedic purposes, or humiliation. Indeed, a contemporary example is tea-bagging in video games (squatting your character over a defeated opponent's face).